# On the Schwarzchild black hole metric:

I've been going through the introduction on Ads/CFT by Horatiu Nastase (https://arxiv.org/abs/0712.0689). In chapter 5, "Black holes and p branes", in Eq. 5.2, it is mentioned the schwarzchild metric to be $ds^2 = -Vdt^2 + \frac{dr^2}{V} + R^2 d\Omega^2$

with $V = ( 1 - \frac{2GM}{r})$; and R = curvature = some constant number.

But the general metric that I see is $ds^2 = -Vdt^2 + \frac{dr^2}{V} + r^2 d\Omega^2$ (ref: https://en.wikipedia.org/wiki/Schwarzschild_metric)

The second metric is with small r (a variable) instead of Big R (a constant) with $d\Omega^2$

I also checked that Einstein equation gets solved with a small r, but with Big R, it doesn't.

But I find it difficult to believe that Horatiu Nastase may be incorrect on this elementary aspect. It has also been cited by many. Does any anyone know if there is something more to the equation given by Nastase?

• It might just be a typo. – user110373 Jun 8 '18 at 4:32
• "I also checked that Einstein equation gets solved with a small r, but with Big R, it doesn't." where did you see this? – Árpád Szendrei Jun 8 '18 at 5:15
• The equation (5.3) following equation (5.2) you cite already uses $r$ instead of $R$, the only difference is that (5.3) is the Newton's approximation of the Schwarzschild metric. Of course during the approximation process $R$ cannot change to $r$. So it's certainly a typo as user110373 already said. – Frederic Thomas Jun 8 '18 at 8:24
• Yes it's a typo (which propagates to the published book!) – Antoine Jun 8 '18 at 11:36